Sec.12.7
Extreme Values and Saddle Points
A local minimum
is a point in the domain that yields the smallest function value in a small region of
the domain about it.
A local maximum
is a point in the domain that yields the largest function value in a small region of
the domain about it.
In dealing with multivariable functions we may have a third type of point in the domain, called a
saddle point,
which appears as a maximum in one direction and a minimum in another direction!
If the extreme points discussed above have the smallest (largest) function value in the whole domain,
we say we have an absolute minimum (maximum).
If
f
has a local maximum or minimum at
(a, b )
and the first order partials exist, then they must equal
zero.
(
,
)0
(
,
xy
f a b
and
f
a b
==
.
An interior point (a, b) of the domain is called a critical point
(stationary point) if both first partials
equal zero or if one of the first partials does not exist.
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 Spring '03
 MECothren
 Calculus, Critical Point, Multivariable Calculus

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